Well that claim could certainly be true. Levels in modern games are too large to fit in memory, so the games have to implement a system where the game only stores the part of the map that is visible in RAM. Whenever you move across the map the game engine has to preload the contents of the area of the map that's about to become visible in advance.
Occasionally the system fails to preload new areas in time and the game "hitches" until the data is retrieved off the hard drive and the data can be processed and rendered. The high HDD access latencies often mean that this process takes long enough to cause noticeable delays. I've actually experienced this myself quite often. SSDs can eliminate hitching because the access latencies are so low that the time it takes for the game to recover from a failure to prefetch data is dramatically reduced.
With online games SSDs won't improve the experience because network latency is a far bigger problem, but they can help for non-online gaming.
A lot of the games data that is loaded into memory is textures. A lot of high end games will stream the textures as it needs them and it's that kind of random access where an SSD would be an improvement over a traditional HDD. It's hard to say how much of an improvement though.
Initial loading times when changing maps for instance would be faster with a good SSD to answer your initial question.